Modeling judgments of large number magnitudes

For the past few months, I’ve been working with David Landy on analyzing and writing up a series of number line judgment experiments. Participants in these experiments see very large numbers (e.g., 783,000,000) and respond by indicating the appropriate location on a number line with known endpoints (e.g., 10^3 and 10^9).

My focus has been on implementing a hierarchical Bayesian model of proportional judgments. At the individual subject level, the model looks (mathematically) like this:

\hat{p}_{j} = \left(\frac{(x_j-R_{i-1})^{\beta}}{(x_j-R_{i-1})^{\beta} + (R_i -x_j)^{\beta}}\right)\left(\frac{K_i-K_{i-1}}{K_m-K_1}\right) + \left(\frac{K_{i-1}-K_1}{K_m-K_1} \right)

Where \hat{p}_{j} is a subject’s judgment of the (proportional) magnitude of x_j, which is the stimulus number on trial j; R_i is an appropriately located reference point (e.g., in the model I’ve been working with, R_i \in \{10^3, 10^6, 10^9\}), and i indicates the smallest reference point greater than x_j; \beta governs the magnitude of ‘cyclic’ responses around (midpoints between) reference points (see below); K_i is a possibly inappropriately located reference point, expressed relative to an arbitrary fixed range; and m is the number of (appropriately and/or inappropriately located) reference points.

For the data I’ve been working with, this can be simplified somewhat. K_m is set equal to 3, and K_1 is set equal to 1. Hence, because there are only three reference points, there is only one free K parameter. (In a data set that we haven’t analyzed yet, m will be 4, K_m will be set equal to 3, K_1 will be set equal to 0, and there will be two free K parameters. But I digress…)

Here’s a picture illustrating predicted proportional judgments for a few different parameter values and for stimuli ranging from 10^3 to 10^9:

The free K parameter corresponds to the fixed reference point at 10^6. This means that if K=1.002, the model predicts correct proportional judgments. (If a stimulus of 10^6 were given, the first term in the model equation above would be 0. The correct proportional judgment for 10^6 is 0.001, so set \hat{p} = 0.001 and solve for the free K parameter, with K_1 = 1 and K_3 = 2, and you get 1.002). The three dashed lines show predicted judgments for K = 1.002.

If \beta = 1 (black lines), the model predicts judgments that are a linear function of the stimuli, while if \beta < 1 (red lines), the model predicts overestimation of lower numbers and underestimation of higher numbers (where lower and higher are defined relative to a point halfway between the fixed reference points), and if \beta > 1 (blue lines), the model predicts underestimation of lower numbers and overestimation of higher numbers.

If K=2, the model predicts an ‘elbow’ in the judgment function, as if the middle fixed reference point at 10^6 were thought to be halfway between 10^3 and 10^9. The K=2 predicted judgments are shown by solid lines, with color indicating \beta values as described above.

Because 10^6 is so much closer to 10^3 than it is to 10^9, when the stimulus numbers are expressed as in the figure above, it’s essentially impossible to see what the K=2 models predict for numbers below 10^6. Here’s the same model predictions as a function of \log_{10} of the stimulus numbers:

This is the basic model. In the data set we’re working on, a number of subjects give judgments that seem to be a mixture of two such models, one with a nice, proper linear K value, and one with a shifted, million-in-the-middle K value. So, I fit a mixture model that, for each subject, assigns each trial’s judgment to a million-in-the-middle component with probability \pi or to a linear component with probability 1-\pi.

The model is also hierarchical, as noted above. This is to say that the free parameters in each individual subject’s mixture model are governed by parent distributions with parameters of their own. As it happens, one of the group-level models is a between-subjects mixture model, which allows for one group of subjects who tend toward linear responses and another who tend toward million-in-the-middle responses. I will write another post about the hierarchical structure and multiple mixtures soon.

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One Response to Modeling judgments of large number magnitudes

  1. Jared Linck says:

    I love that you described this as the “basic” model. If I hadn’t already talked about these data with you a half dozen times, there is no way that I would have understood everything you described (and I don’t know that I do now!).

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